An Incremental Finite Element Formulation for the Static Response of Non-Uniform Transversally Functionally Graded I-Beams

Tapered doubly symmetric I-beams are widely used across structural, civil, and aeronautical engineering due to their excellent strength-to-weight ratio, stability, and material efficiency. With recent advancements in manufacturing and the growing application of functionally graded materials (FGMs), understanding how material heterogeneity interacts with non-uniform beam geometry has become increasingly important. This paper presents a straightforward incremental finite element approach tailored for the static analysis of transversally functionally graded (TFG) tapered I-beams based on Euler–Bernoulli beam theory. To capture the simultaneous variations in geometry and material properties smoothly, the proposed formulation discretizes non-uniform beams into a series of smaller, kinematically equivalent segments with uniform cross-sections. The validity, numerical stability, and convergence of this approach are evaluated through three detailed benchmark examples. Beyond validating the technique, we conduct a comprehensive parametric study to examine how material property profiles, tapering ratios, and end boundary conditions jointly influence the beam's structural response. The material variation across the cross-section is governed by a power-law distribution. Our results reveal that higher values of the power-law exponent reduce the effective flexural stiffness of the beam, leading to noticeably larger deflections under load. Similarly, lower tapering parameters increase the overall compliance of the member, resulting in higher flexibility across various boundary configurations. By offering a clear and practical numerical framework, this study provides valuable insights into the design and performance optimization of non-uniform functionally graded structural components in modern engineering applications.